Once the matrix \(\mathbf {M}_l^i\)  in (REF ) at the top of next page is derived, the estimation of the transmitted message can be obtained.
All items in \(\mathbf {M}_l^i\)  can be calculated directly, except for \(\Sigma _l^i\) . According to the property that \((\mathop \diamond \limits _{j=1}^{d}\mathbf {A}^j)^T(\mathop \diamond \limits _{j=1}^{d}\mathbf {A}^j)^*=\mathop \odot \limits _{j=1}^{d}\mathbf {A}^{j,T}\mathbf {A}^{j,*}\)  [1]}, the expectation of the coupled and individual factors terms in \((\Sigma _l^i)^{-1}\)  of (REF ) reduces to
\(&\mathbb {E}\left[\left(\!\mathop \diamond \limits _{j=1,j\ne i}^{d}\mathbf {X}_{l}^j\mathop \diamond \limits \mathbf {P}_l\mathbf {G}\!\right)^T\!\left(\!\mathop \diamond \limits _{j=1,j\ne i}^{d}\mathbf {X}_{l}^j\mathop \diamond \limits \mathbf {P}_l\mathbf {G}\!\right)^*\right]\nonumber \\&=\mathcal {D}\left[\mathop \odot \limits _{j=1,j\ne i}^{d}\mathbb {E}\left[\mathbf {X}_{l}^{j,T}\mathbf {X}_{l}^{j,*} \right]\odot \mathbb {E}\left[\mathbf {G}^T\mathbf {P}_l^T\mathbf {P}_l\mathbf {G}^*\right]\right],\) 
